| L(s) = 1 | + 2-s + 4·5-s − 7-s + 8-s − 2·9-s + 4·10-s + 2·11-s + 2·13-s − 14-s − 16-s + 6·17-s − 2·18-s − 3·19-s + 2·22-s + 6·25-s + 2·26-s + 3·27-s + 2·29-s + 10·31-s − 6·32-s + 6·34-s − 4·35-s − 8·37-s − 3·38-s + 4·40-s − 4·41-s − 6·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·5-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.471·18-s − 0.688·19-s + 0.426·22-s + 6/5·25-s + 0.392·26-s + 0.577·27-s + 0.371·29-s + 1.79·31-s − 1.06·32-s + 1.02·34-s − 0.676·35-s − 1.31·37-s − 0.486·38-s + 0.632·40-s − 0.624·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100293 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100293 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.948530124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.948530124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 101 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 331 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 5 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 6 T + T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 148 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 110 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 127 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 101 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 76 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 97 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.9276693546, −13.7594567335, −13.3424836010, −12.9334973175, −12.2864943326, −12.0741582574, −11.5493245751, −10.7875379511, −10.4329332878, −10.1610641236, −9.50115534256, −9.24530657808, −8.63060568159, −8.22066652617, −7.53438197513, −6.67277670784, −6.38237267810, −6.06148989040, −5.35032885515, −5.03313787499, −4.33893291568, −3.50611719070, −2.97190383934, −2.09148230339, −1.33424885817,
1.33424885817, 2.09148230339, 2.97190383934, 3.50611719070, 4.33893291568, 5.03313787499, 5.35032885515, 6.06148989040, 6.38237267810, 6.67277670784, 7.53438197513, 8.22066652617, 8.63060568159, 9.24530657808, 9.50115534256, 10.1610641236, 10.4329332878, 10.7875379511, 11.5493245751, 12.0741582574, 12.2864943326, 12.9334973175, 13.3424836010, 13.7594567335, 13.9276693546
